Download 73 013601 (2006)

PHYSICAL REVIEW A 73, 013601 共2006兲
Transition to instability in a periodically kicked Bose-Einstein condensate on a ring
1
Jie Liu,1,2,* Chuanwei Zhang,2,3,* Mark G. Raizen,2,3 and Qian Niu2
Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
2
Department of Physics, The University of Texas, Austin, Texas 78712-1081, USA
3
Center for Nonlinear Dynamics, The University of Texas, Austin, Texas 78712-1081, USA
共Received 29 August 2005; published 3 January 2006兲
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the
quantum kicked rotor, where the nonlinearity stems from the mean-field interactions between the condensed
atoms. For weak interactions, periodic motion 共antiresonance兲 becomes quasiperiodic 共quantum beating兲 but
remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes
chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of
noncondensed atoms. In the stable regime, the system remains predominantly in the two lowest energy states
and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and
discuss the route to instability. We numerically explore a parameter regime for the occurrence of instability and
reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to
higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for
dynamically localized states 共essentially quasiperiodic motions兲, where stability remains for weak interactions
but is destroyed by strong interactions.
DOI: 10.1103/PhysRevA.73.013601
PACS number共s兲: 03.75.Kk, 05.45.⫺a, 42.50.Vk
I. INTRODUCTION
The ␦-kicked rotor is a textbook paradigm for the study of
classical and quantum chaos 关1兴. In the classical regime, increasing kick strengths destroy regular periodic or quasiperiodic motions of the rotor and lead to the transition to chaotic
motions, characterized by diffusive growth in the kinetic energy. In quantum mechanics, chaos is no longer possible because of the linearity of the Schrödinger equation and the
motion becomes periodic 共anti-resonance兲, quasiperiodic
共dynamical localization兲, or resonant 共quantum resonance兲
关2,3兴. Experimental study of these quantum phenomena have
been done with ultracold atoms in periodically pulsed optical
lattices 关4兴. However, most of the previous investigations
have been focused on single-particle systems and the effects
of interaction between particles have not received much attention 关5,6兴.
In recent years, the realization of Bose-Einstein condensation 共BEC兲 关7兴 of dilute gases has opened interesting opportunities for studying dynamical systems in the presence of
many-body interactions. One cannot only prepare initial
states with unprecedented precision and pureness, but one
also has the freedom of introducing interactions between the
particles in a controlled manner. A natural question to ask is
how the physics of the quantum kicked rotor is modified by
the interactions. In the mean-field approximation, manybody interactions in BEC are represented by adding a nonlinear term in the Schrödinger equation 关8兴 共such a nonlinear
Schrödinger equation also appears in the context of nonlinear
optics 关9兴兲. This nonlinearity makes it possible to bring chaos
back into the system, leading to instability 共in the sense of
exponential sensitivity to initial conditions兲 of the conden-
*These authors contributed equally to this work.
1050-2947/2006/73共1兲/013601共10兲/$23.00
sate wave function 关10兴. The onset of instability of the condensate can cause rapid proliferation of thermal particles 关11兴
that can be observed in experiments 关12兴. It is therefore important to understand the route to chaos with increasing interactions. This problem has recently been studied for the
kicked BEC in a harmonic oscillator 关6兴.
In Ref. 关13兴, we have investigated the quantum dynamics
of a BEC with repulsive interaction that is confined on a ring
and kicked periodically. This system is a nonlinear generalization of the quantum kicked rotor. From the point of view
of dynamical theory, the kicked rotor is more generic than
the kicked harmonic oscillator, because it is a typical low
dimensional system that obeys the KAM theorem, while the
kicked harmonic oscillator is known to be a special degenerate system out of the framework of the KAM theorem 关14兴.
It is very interesting to understand how both quantum mechanics and mean-field interaction affect the dynamics of
such a generic system.
In this paper, we extend the results of Ref. 关13兴, including
a more detailed analysis of the model considered there as
well as different phenonmena. We will focus our attention on
the relatively simpler case of quantum antiresonance, and
show how the state is driven towards chaos or instability by
the mean-field interaction. The paper is orgnaized as follows:
Section II lays out our physical model and its experimental
realization. Section III is devoted to the case of weak interactions between atoms. We find that weak interactions make
the periodic motion 共antiresonance兲 quasiperiodic in the form
of quantum beating. However, the system remains predominantly in the lowest two energy levels and analytic expressions for the beating frequencies are obtained by mapping
the system onto a spin model. Through varying the kick period, we find that the phenomenon of antiresonance may be
recovered even in the presence of interactions. The decoherence effects due to thermal noise are discussed. Section IV is
devoted to the case of strong interactions. It is found that
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©2006 The American Physical Society
PHYSICAL REVIEW A 73, 013601 共2006兲
LIU et al.
there exists a critical strength of interactions beyond which
quasiperiodic motion 共quantum beating兲 is destroyed, resulting in a transition to instability of the condensate characterized by an exponential growth in the number of noncondensed atoms. Universal critical behavior for the transition is
found. We show that the occurrence of instability corresponds to the process of Arnold diffusion, through which the
state can penetrate through the KAM tori and escape to highenergy levels 关15兴. We study nonlinear effects on dynamically localized states that may be regarded as quasiperiodic
关16兴. Similar results are obtained in that localization remians
for sufficiently weak interactions but become unstable beyond a critical strength of interactions. Section V consists of
conclusions.
II. PHYSICAL MODEL: KICKED BEC ON A RING
Although the results obtained in this paper are common
properties of systems whose dynamics are governed by the
nonlinear Schrödinger equation, we choose to present them
here for a concrete physical model: a kicked BEC confined
on a ring trap, where the physical meanings of the results are
easy to understand. The description of the dynamics of this
system may be divided into two parts: condensed atoms and
noncondensed atoms.
A. Dynamics of condensed atoms: Gross-Pitaevskii equation
Consider condensed atoms confined in a toroidal trap of
radius R and thickness r, where r Ⰶ R so that lateral motion is
negligible and the system is essentially one-dimensional
关17兴. The dynamics of the BEC is described by the dimensionless nonlinear Gross-Pitaveskii 共GP兲 equation,
冉
冊
1 ⳵2
⳵
+ g兩␺兩2 + K共cos␪兲␦t共T兲 ␺ ,
i ␺= −
2 ⳵␪2
⳵t
be adjusted using a magnetic-field-dependent Feshbach resonance or the variation of the number of atoms 关21兴.
B. Dynamics of noncondensed atoms: Bogoliubov equation
The deviation from the condensate wave function is described by Bogoliubov equation that is obtained from a linearization around the GP equation 关11兴. In Castin and Dum’s
formalism, the mean number of noncondensed atoms at zero
⬁
具vk共t兲 兩 vk共t兲典, where
temperature is described by 具␦N̂共t兲典 = 兺k=1
vk共t兲 are governed by
i
冊冉 冊
uk
vk
,
共2兲
where H1 = p̂2 / 2 + g兩␺兩2 − ␮ + gQ兩␺兩2Q + K共cos ␪兲␦t共T兲, H2
= gQ␺2Q*, ␮ is the chemical potential of the ground state, ␺
is the ground state of the GP equation, and the projection
operators Q are given by Q = 1 − 兩␺典具␺兩.
The number of noncondensed atoms describes the deviation from the condensate wave function and its growth rate
characterizes the stability of the condensate. If the motion of
the condensate is stable, the number of noncondensed atoms
grows at most polynomially with time and no fast depletion
of the condensate is expected. In contrast, if the motion of
the condensate is chaotic, the number of noncondensed atoms diverges exponentially with time and the condensate
may be destroyed in a short time. Therefore the rate of
growth of the noncondensed atoms number is similar to the
Lyapunov exponent for the divergence of trajectories in
phase space for classical systems 关6兴.
III. WEAK INTERACTIONS: ANTIRESONANCE AND
QUANTUM BEATING
共1兲
where g = 8NaR / r2 is the scaled strength of nonlinear interaction, N is the number of atoms, a is the s-wave scattering
length, K is the kick strength, ␦t共T兲 represents 兺n␦共t − nT兲, T
is the kick period, and ␪ denotes the azimuthal angle. The
length and the energy are measured in units R and ប2 / mR2,
respectively. The wave function ␺共␪ , t兲 has the normalization
兰20␲兩␺兩2d␪ = 1 and satisfies periodic boundary condition
␺共␪ , t兲 = ␺共␪ + 2␲ , t兲.
Experimentally, the ring-shape potential may be realized
using two two-dimensional 共2D兲 circular “optical billiards”
with the lateral dimension being confined by two plane optical billiards 关18兴, or optical-dipole traps produced by reddetuned Laguerre-Gaussian laser beams of varying azimuthal
mode index 关19兴. The ␦ kick may be realized by adding
potential points along the ring with an off-resonant laser 关4兴,
or by illuminating the BEC with a periodically pulsed
strongly detuned running wave laser in the lateral direction
whose intensity I is engineered to I = I0 + I1x / R 关20兴. In the
experiment, we have the freedom to replace the periodic
modulation cos共␪兲 of the kick potential with cos共j␪兲, where j
is an integer. Such replacement revises the scaled interaction
constant g and kicked strength K, but will not affect the
results obtained in the paper. The interaction strength g may
冉冊冉
H1
H2
d uk
=
*
dt vk
− H2 − H*1
In this section, we focus our attention on the case of quantum antiresonance, and show how weak interactions between
atoms modify the dynamics of the condensate. Quantum antiresonance is a single-particle phenomenon characterized by
periodic recurrence between two different states, and its dynamics may be described by Eq. 共1兲 with parameters g = 0
and T = 2␲ 关3兴.
A. Quantum beating
In a noninteracting gas, the energy of each particle oscillates between two values because of the periodic recurrence
of the quantum states. In the presence of interactions, singleparticle energy loses its meaning and we may evaluate the
mean energy of each particle
E共t兲 =
冕
2␲
0
冉
d␪␺* −
冊
1 ⳵2 1
+ g兩␺兩2 ␺ .
2 ⳵␪2 2
共3兲
To determine the evolution of the energy, we numerically
integrate Eq. 共1兲 over a time span of 100 kicks, using a
split-operator method 关22兴, with the initial wave function ␺
being the ground state ␺共␪ , 0兲 = 1 / 冑2␲. After each kick, the
energy E共t兲 is calculated and plotted as shown in Fig. 1.
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FIG. 3. Plots of beat and oscillation frequencies versus the interaction strength 共a兲 and kick strength 共b兲, where the scatters are
the results from numerical simulation using the GP equation and
lines from analytic expressions Eqs. 共15兲 and 共16兲.
FIG. 1. Plots of average energy E共t兲 versus the number of kicks
t for three values of g. The kick strength K = 0.8.
with respect to the interaction strength g. More interestingly,
the two frequencies satisfy a conservation relation
In the case of noninteraction 关Fig. 1共a兲, g = 0兴, we see that
the energy E共t兲 oscillates between two values and the oscillation period is 2T, indicating the periodic recurrence between two states 共antiresonance兲.
The energy oscillation with weak interaction 共g = 0.1兲 in
Fig. 1共b兲 shows a remarkable difference from that for the
noninteraction case. We see that the amplitude of the oscillation decreases gradually to zero and then revives, similar to
the phenomenon of beating in classical waves. Clearly, it is
the interactions between atoms in BEC that modulate the
energy oscillation and produce the phenomena of quantum
beating. As we know from classical waves, there must be
two frequencies, oscillation and beat, to create a beating.
These two frequencies are clearly seen in Fig. 2 that is obtained through Fourier transform of the energy evolutions in
Fig. 1. For the noninteraction case 关Fig. 2共a兲兴, there is only
the oscillation frequency f osc = 0.5/ T, corresponding to one
oscillation in two kicks. The interactions between atoms develop a new beat frequency f beat, as well as modify the oscillation frequency f osc, as shown in Fig. 2共b兲.
In Fig. 3, these two frequencies are plotted with respect to
different interacting constant g and kick strength K. We see
that both beat and oscillation frequencies vary nearly linearly
f osc + f beat/2 = 1/2.
For a strong interaction 关Fig. 1共c兲兴, i.e., g 艌 1.96, we find
that the energy’s evolution demonstrates an irregular pattern,
clearly indicating the collapse of the quasiperiodic motion
and the occurrence of instability. The corresponding Fourier
transformation of the energy evolution 关Fig. 2共c兲兴 has no
sharp peak. This transition to instability will be discussed in
Sec. IV in details.
B. Spin model
The phenomena of quantum beating can be understood by
considering a two-mode approximation 关23兴 to the GP equation. In this approximation, condensed atoms can only effectively populate the two lowest second quantized energy
modes. The validity of this two-mode model is justified by
the following facts which are observed in the numerical
simulation. First, the total energy of the condensate is quite
small so that we can neglect the population at high-energy
modes and keep only the states with quantized momentums 0
and ±1. Second, the total momentum of the condensate is
conserved during the evolution. Therefore the populations of
the states with momentum ±1 are same if the initial momentum of the condensate is zero 共the ground state兲.
Here we consider a quantum approach of this two-mode
model, which yields an effective spin Hamiltonian in the
mean-field approximation. By considering the conservation
of parity we may write the Boson creation operator for the
condensate as
1
†
†
␺ˆ † =
冑2␲ 共â + 2b̂ cos ␪兲,
FIG. 2. Fourier transformation of the energy evolutions in Fig.
1. The unit for the frequency is 1 / T.
共4兲
共5兲
where ↠and b̂† are the creation operators for the ground
state and the first excited states, satisfying the commutation
relation 关â , â†兴 = 1, 关b̂ , b̂†兴 = 1, and particle number conservation â†â + b̂†b̂ = N.
Substituting Eq. 共5兲 into the many-body Hamiltonian of
the system
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PHYSICAL REVIEW A 73, 013601 共2006兲
LIU et al.
冕 冋冉
2␲
Ĥ =
0
冊
1 ⳵2 ˆ
g ˆ†ˆ†ˆ ˆ
d␪ ␺ˆ † −
␺ ␺ ␺␺
2 ␺+
2 ⳵␪
2N
Ṡz =
册
+ ␺ˆ †„K共cos ␪兲␦T共t兲…␺ˆ ,
共6兲
冉
H=−
冊
共7兲
in terms of the Bloch representation by defining the angular
momentum operators,
L̂x =
â†b̂ + âb̂†
,
2
L̂y =
â†b̂ − âb̂†
,
2i
L̂z =
â†â − b̂†b̂
,
2
dL̂z
g
=
兵L̂y,L̂x其 + 冑2KL̂y␦T共t兲,
dt ␲N
共9兲
where 兵L̂i , L̂ j其 = L̂iL̂ j + L̂ jL̂i.
The mean-field equations for the first-order expectation
values 具L̂i典 of the angular momentum operators are obtained
Ṡz = 冑2KSy␦T共t兲.
products of 具L̂i典 and 具L̂ j典, that is, 具L̂iL̂ j典 = 具L̂i典具L̂ j典 关24兴. Defining the single-particle Bloch vector
2
Sជ = 共Sx,Sy,Sz兲 = 共具L̂x典,具L̂y典,具L̂z典兲,
N
we obtain the nonlinear Bloch equations
冊
1
g
g
−
Sz S y ,
+
2 8␲ 8␲
冊
1
7g
g
−
−
Sz Sx − 冑2KSz␦T共t兲,
2 8␲ 8␲
共13兲
We see that the evolution between two consecutive kicks is
simply an angle ␲ rotation about the z axis, which yields the
spin transformation Sx → −Sx, Sy → −Sy. The spin initially directing to north pole 关Sជ = 共0 , 0 , 1兲兴 stays there for time duration T, then the first kick rotates the spin by an angle 冑2K
about the x axis and now Sជ = (0 , −sin共冑2K兲 , cos共冑2K兲). The
following free evolution rotates the spin to Sជ
= (0 , sin共冑2K兲 , cos共冑2K兲). Then, the second kick will drive
the spin back to north pole through another rotation of 冑2K
about the x axis. With this the spin’s motion is two kick
period recurrence and quantum antiresonance occurs.
The motion of the spin is more complicated with interactions. The spin components at x̂ and ŷ directions can be
written as Sx = 冑1 − Sz2 cos ␣ and Sy = −冑1 − Sz2 sin ␣ in terms
of population difference Sz and relative phase ␣, which
yields the relation
Ṡx = ␣˙ Sy −
by approximating second-order expectation values 具L̂iL̂ j典 as
冉
共12兲
From the definition of the Bloch vector, we see that Sz corresponds to the population difference and −arctan共Sy / Sx兲 corresponds to the relative phase ␣ between the two modes.
This Hamiltonian is similar to a kicked top model 关25兴, but
here the evolution between two kicks is more complicated.
With the spin model, we can readily study the dynamics
of the system. For the noninteraction case 共g = 0兲, The Bloch
equations 共11兲 become
共8兲
7g
L̂x g共N − 1兲
dL̂y
=− −
L̂x −
兵L̂z,L̂x其 − 冑2KL̂z␦T共t兲,
dt
2
8␲N
8␲N
Ṡy = −
冊
1
Ṡy = − Sx − 冑2KSz␦T共t兲,
2
g
dL̂x L̂y g共N − 1兲
=
+
L̂y −
兵L̂y,L̂z其,
dt
2
4N
8␲N
冉
冉
Sz
g
Sz S2
+
S2x − + z + 冑2KSx␦t共T兲.
8
2 2␲
4
1
Ṡx = Sy ,
2
where we have discarded all c-number terms.
The Heisenberg equations of motion for the three angular
momentum operators reads
Ṡx =
共11兲
where we have used N Ⰷ 1. The mean-field Hamiltonian in
the spin representation reads
we obtain a quantized two-mode Hamiltonian
1
1
1
g
4L̂2x − 共N − 1兲L̂z + L̂z2 + 冑2KL̂x␦T共t兲
Ĥ = − L̂z +
2
2
2
4␲N
g
SySx + 冑2KSy␦T共t兲,
␲
g
SzSy cos2 ␣
␲
during the free evolution. Comparing this equation with the
Bloch equation 共11兲, we obtain the equation of motion for the
relative phase,
共10兲
␣˙ =
冉
冊
1
3
g
g
+
+
cos 2␣ + Sz .
2 8␲ 2␲
4
共14兲
We see the motion between two consecutive kicks is approximately described by a rotation of ␲ + g共1 + 3Sz兲 / 4 about
the z axis. Compared with the noninteraction case, the meanfield interaction leads to an additional phase shift g共1
+ 3Sz兲 / 4. This phase shift results in a deviation of the spin
from Sx = 0 plane at time 2T−, i.e., the moment just before the
second kick. As a result, the second kick cannot drive the
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TRANSITION TO INSTABILITY IN A PERIODICALLY…
FIG. 4. Periodic stroboscopic plots of the projection of spin on
the Sz = 0 plane. The thick line and dots correspond to the orbits
with initial spin Sជ = 共0 , 0 , 1兲. K = 0.8 共a兲 g = 0.0; 共b兲 g = 0.1.
spin back to its initial position and quantum antiresonance is
absent. However, the phase shift will be accumulated in future evolution and the spin may reach the Sx = 0 plane at a
certain time mT− 共beat period兲 when the total accumulated
phase shift is ␲ / 2. Then the next kick will drive the spin
back to north pole by applying an angle 冑2K rotation about
the x axis.
The above picture is confirmed by our numerical solution
of the spin Hamiltonian with the fourth-order Runge-Kutta
method 关26兴. In Fig. 4, we plot the phase portraits of the spin
evolution by choosing different initial conditions of the
population difference Sz and relative phase ␣. Just after each
kick three spin components Si 共i = x , y , z兲 are determined and
their projections on the Sz = 0 plane are plotted. For the noninteraction case 关Fig. 4共a兲兴, we see that the motion of the spin
is an oscillation between the north pole A and another point
B, indicating the occurrence of quantum antiresonance. The
interaction between atoms changes the phase portraits dramatically 关Fig. 4共b兲兴. Around the north pole, a fixed point
surrounded by periodic elliptic orbits appears. The two-point
oscillation is shifted slowly and forms a continuous and
closed orbit, representing the phenomenon of quantum beating.
In Fig. 5 we see that the relative phase at the moment just
before the even kicks increases almost linearly and reaches
2␲ in a beat period. The slope of the increment reads, ␥RP
= 关␣共4T−兲 − ␣共2T−兲兴 / 2, which can be deduced analytically.
With this and through a lengthy deduction, we obtain an
analytic expression for the beat frequency to first order in g,
f beat ⬇
g
关1 + 3 cos共冑2K兲兴.
4␲
共15兲
FIG. 6. 共a兲 Plot of population difference versus the number of
kicks t, where g = 0.1, K = 0.8. 共b兲 Details of 共a兲 at the middle of a
beat period. The population difference decreases in two consecutive
kicks.
In Fig. 6, we plot the evolution of population difference
and the phenomenon of quantum beating is clearly seen in
the spin model. Notice that there is a one-peak miss of the
oscillation at the middle of one period because the population difference decreases in two consecutive kicks. Taking
account of this missed peak, we obtain the oscillation frequency
f osc ⬅
Ntotal − Nmiss 1 1
= − f beat ,
2 2
2Ntotal
共16兲
where Ntotal and Nmiss are total and missed numbers of peaks,
respectively.
The analytical expressions Eqs. 共15兲 and 共16兲 of the beat
and oscillation frequencies are in very good agreement with
the numerical results obtained from the GP equation, as
shown in Fig. 3. Therefore the beating provides a method to
measure interaction strength in an experiment.
C. Antiresonance with interactions
In the spin model, we see that it is the additional phase
shift originating from weak interactions that destroys the two
kick period recurrence of the antiresonance and leads to the
phenomenon of quantum beating. Therefore we will still be
able to observe the quantum antiresonance even in the presence of interactions if the additional phase shift may be compensated. Actually, the additional phase shift can be canceled
by varying the kick period T so that the relative phase ␣ only
changes ␲ between two consecutive kicks. Using Eq. 共14兲,
we find that the new kick period for antiresonance in the
presence of interactions may be approximated as
TAR ⬇
8␲2
4␲ + g + 3g cos共冑2K兲
.
共17兲
In Fig. 7, we plot the evolution of the average energy E共t兲
with the new kick period TAR. We see that the energy oscillates between two values and the oscillation period is 2TAR,
clearly indicating the recovery of the antiresonance.
D. Decoherence due to thermal noise
FIG. 5. 共a兲 Plots of relative phase versus the number of kicks t,
where g = 0.1, K = 0.8. 共b兲 Schematic plot of the phase shift. nT−共+兲
represents the moment just before 共after兲 the nth kick.
In realistic experiments, the decoherence effects always
exist. Generally, decoherence originates in the coupling to a
bath of unobserved degree of freedom, or the interparticle
entanglement process 关27,28兴. The main source of decoher-
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LIU et al.
FIG. 7. Plots of average energy E共t兲 versus the number of kicks
t for K = 0.1, g = 0.1. TAR is determined through Eq. 共17兲.
ence in a BEC is the thermal cloud of particles surrounding
the condensate. Thermal particles scattering off the condensate will cause phase diffusion at a rate ⌫ proportional to the
thermal cloud temperature. For internal states not entangled
with the condensate spatial state, ⌫ may be as low as 10−5 Hz
under the coldest experimental condition 关24兴, it may reach
2 Hz 关29兴 or higher 关30兴 depending on the practical experimental situation.
Modeling decoherence by fully include the quantum effects requires sophisticated theoretical studies that nontrivially include noncondensate atoms. In the experiments on
BECs of dilute atomic gases trapped atoms are evaporatively
cooled and they continuously exchange particles with their
environment. Thus standard approaches of quantum optics
for open systems involving quantum kinetic master equations
seem especially natural for treating BECs 关31兴,
␳˙ = i关␳,H兴 −
⌫
兺 †l̂†l̂,关l̂†l̂, ␳兴‡,
2 l=a,b
共18兲
where ␳ is the N-body density operator.
Different approaches can be used to solve the Master
equations 关24兴, including 共i兲 the mean-field treatment, 共ii兲 the
mean-field solution plus first-order quantum fluctuation, and
共iii兲 the exact quantum solution. For our case of antiresonance and beating the mean-field solutions are stable so that
the quantum breaking time is expected to be long enough
共⬃N兲. It implies that the mean-field solution will follow the
exact quantum solution for a long time. So, we adopt the
simple mean-field treatment in our simulation. From the
mean-field viewpoint, the decoherence term in Eq. 共18兲 introduces a ⌫ transversal relaxation term into the mean-field
equations of motion,
冉
冉
冊
冊
1
7g
g
−
Sz Sx − 冑2KSz␦T共t兲 − ⌫Sy ,
−
2 8␲ 8␲
Ṡz =
g
SySx + 冑2KSy␦T共t兲.
␲
In Fig. 8, we plot the evolution of the population difference Sz for different decoherence constant ⌫. We see that the
phenomenon of quantum beating is destroyed by strong thermal noise 关Fig. 8共b兲兴, while it survives in weak noise 关Fig.
8共a兲兴. For large decoherence constant, the population difference Sz decays to 0 exponentially and the characteristic time
is the just the decoherence time ␶ = 1 / ⌫. Therefore the decoherence time ␶ must be much larger than the beat period
2␲ / f beat to observe the phenomenon of quantum beating,
which yields
2␲⌫/g Ⰶ
1
关1 + 3 cos共冑2K兲兴.
4␲
共20兲
In the case of K = 0.8, Eq. 共20兲 gives an estimation 2␲⌫ / g
Ⰶ 0.2, which agrees with the numerical results shown in Fig.
8.
If we suppose our ring shape parameters of r = 5 ␮m, R
= 50 ␮m 关32兴 and use the typical data of 87Rb, a = 5.7 nm, the
atom number is 104, the threshold rate approximates to a few
tens of Hz, that is controllable in a practical situation.
IV. STRONG INTERACTIONS: TRANSITION TO
INSTABILITY
A. Characterization of the instability: Bogoliubov excitation
1
g
g
−
Sz Sy − ⌫Sx ,
+
Ṡx =
2 8␲ 8␲
Ṡy = −
FIG. 8. Plots of population difference with respect to the number of kicks in the presence of decoherence. K = 0.8, g = 0.2. Thin
lines correspond to ⌫ = 0. Thick lines correspond to 共a兲 2␲⌫ / g
= 0.001, 共b兲 2␲⌫ / g = 1.
共19兲
Tuning the interaction strength still larger means enhancing further the nonlinearity of the system. From our general
understanding of nonlinear systems, we expect that the solution will be driven towards chaos, in the sense of exponential
sensitivity to initial condition and random evolution in the
temporal domain. The latter character has been clearly displayed by the irregular pattern of the energy evolution in Fig.
1共c兲. On the other hand, the onset of instability 共or chaotic
motion兲 of the condensate is accompanied with the rapid
proliferation of thermal particles. Within the formalism of
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TRANSITION TO INSTABILITY IN A PERIODICALLY…
FIG. 9. 共a兲 Semilog plot of the mean number of noncondensed
atoms versus the number of kicks t. The thicker lines are fitting
functions. K = 0.8, g = 0.1 共dashed line, fitting function 0.0003t1.3兲,
g = 1.5 共dotted line, fitting function 0.0011t2兲, g = 2.0 关dash-dotted
line, fitting function 0.32 exp共0.1t兲兴. The inset shows the interaction
dependence of the growth rate. The scatters are from numerical
simulation and the solid line is the fitting function 0.33共g
− 1.96兲1/2. 共b兲 Phase diagram of the transition to instability.
Castin and Dum 关11兴 described in Sec. II, the growth of the
number of the noncondensed atom will be exponential, similar to the exponential divergence of nearby trajectories in
phase space of classical system. The growth rate of the noncondensed atoms is similar to the Lyapunov exponent, turning from zero to nonzero as instability occurs.
We numerically integrate Bogoliubov equation 共2兲 for the
uk, vk pairs over a time span of 100 kicks, using a split
operator method, parallel to numerical integration of the GP
equation 共1兲. The initial conditions
冉 冊 冉 冊
uk共0兲
vk共0兲
=
1 ␨ + ␨−1 ik␪
e
2 ␨ − ␨−1
共21兲
for initial ground-state wave function ␺共␪兲 = 1 / 冑2␲, are obtained by diagonalizing the linear operator in Eq. 共2兲 关33兴,
where ␨ = 关共k2 / 2兲 / 共k2 / 2 + 2g兩␺兩2兲兴1/4.
After each kick the mean number of noncondensed atoms
is calculated and plotted versus time in Fig. 9共a兲. We find that
there exists a critical value for the interaction strength, i.e.,
gc = 1.96, above which the mean number of noncondensed
atoms increases exponentially, indicating the instability of
BEC. Below the critical point, the mean number of noncondensed atoms increases polynomially. As the nonlinear parameter crosses over the critical point, the growth rate turns
from zero to nonzero, following a square-root law 关inset in
Fig. 9共a兲兴. This scaling law may be universal for Bogoliubov
excitation as confirmed by recent experiments 关12兴.
The critical value of the interaction strength depends on
the kick strength. For very small kick strength, the critical
interaction is expected to be large, because the ground state
of the ring-shape BEC with repulsive interaction is dynamically stable 关34兴. For large kick strength, to induce chaos, the
interaction strength must be large enough to compete with
the external kick potential. So, in the parameter plane of
共g , k兲, the boundary of instability shows a “U”-type curve
关Fig. 9共b兲兴.
Across the critical point, the density profiles of both condensed and noncondensed atoms change dramatically. In Fig.
FIG. 10. Plots of condensate and noncondensate densities,
where K = 0.8. 共a兲, 共b兲 g = 0.1; 共c兲, 共d兲 g = 2.0.
10, we plot the temporal evolution of the density distributions of condensed atoms as well as noncondensed atoms. In
the stable regime, the condensate density oscillates regularly
with time and shows a clear beating pattern 关Fig. 10共a兲兴,
whereas the density of the noncondensed atoms grows
slowly and shows main peaks around ␪ = ± ␲ and 0, besides
some small oscillations 关Fig. 10共b兲兴. In the unstable regime,
the temporal oscillation of the condensate density is irregular
关Fig. 10共c兲兴, whereas the density of noncondensed atoms
grows explosively with the main concentration peaks at ␪
= ± ␲ / 2 where the gradient density of the condensed part is
maximum 关Fig. 10共d兲兴. Moreover, our numerical explorations show that the cos2 ␪ mode 关Fig. 10共b兲兴 dominates the
density distribution of the noncondensed atoms as the interaction strength is less than 1.8. Thereafter, the sin2 ␪ mode
grows while cos2 ␪ mode decays, and finally sin2 ␪ mode
become dominating in the density distribution of noncondensed atoms above the transition point 关Fig. 10共d兲兴. Since
the density distribution can be measured in experiment, this
effect can be used to identify the transition to instability.
B. Arnold diffusion
We have seen that strong interactions destroy the beating
solution of the GP equation and the motion of the condensate
becomes chaotic, characterized by exponential growth in the
number of noncondensed atoms. The remaining question is
how the motion of condensate is driven to chaos, that is, the
route of the transition to instability.
The transition to chaos for the motion of the condensate
can be clearly seen in the periodic stroboscopic plots of the
trajectories for both two-mode approximation 关Figs.
11共a兲–11共d兲兴 and four-mode approximation 关Figs. 11共e兲 and
11共f兲; it is exact for the interaction region we consider in Fig.
11兴 to the original GP equation 共1兲. The solution oscillates
between two points A and B 共point C is identical to A in the
the spin model兲 for the noninteraction case, and forms a
closed path in the phase space for the weak interaction 关Fig.
11共a兲兴. Note that compared with the two-mode calculation,
the effective interaction strength in the four-mode approxi-
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LIU et al.
chastic region to another are still blocked by KAM tori for
the two-mode system. In Fig. 11共d兲, the trajectory with initial
condition Sz = 1, ␣ = 0 is closer to the chaotic region.
The above discussion is based on two-mode approximation; actually, the solution is coupled with other modes of
higher energy states. For small interaction, this coupling is
negligible and the four-mode simulation gives the same results as seen in Figs. 11共a兲, 11共b兲, 11共e兲, and 11共f兲. For large
interaction, this coupling is important and our system is essentially high dimensional 共d ⬎ 2兲. One important character
of a high-dimensional dynamical system is that KAM tori 共d
dimension兲 cannot separate phase space 共2d dimension兲 and
the whole chaotic region is interconnected. If a trajectory lies
in a chaotic region it can circumvent KAM tori and diffuse to
higher energy states through Arnold diffusion. This process
is clearly demonstrated in Figs. 11共g兲 and 11共h兲. We see that
the trajectory diffuses along the separatrix layers, circumvents the KAM tori, and finally spreads over whole phase
space 关Fig. 11共h兲, g = 2.5兴. We also calculate the diffusion
coefficient
DE =
FIG. 11. Periodic stroboscopic plots of population difference
with respect to the relative phase between the first two modes. K
= 0.8. 共a兲–共d兲 the two-mode model, where 共a兲 g = 0.1, circle dots
corresponds to g = 0; 共b兲 g = 1.5; 共c兲 g = 1.9; 共d兲 g = 2.0; The thicker
line and larger dots on the phase portraits represent the trajectories
with initial conditions Sz = 1, ␣ = 0. 共e兲–共h兲 the four-mode approximation. The portrait is the projection on the first two modes of the
trajectory with initial condition Sz = 1, ␣ = 0. 共e兲 g = 0.1; 共f兲 g = 1.6;
共g兲 g = 2.2; 共h兲 g = 2.5.
mation is rescaled, to give the comparable pattern in the
phase space.
With increasing interaction strength, the stable quasiperiodic orbits in Fig. 11共b兲 bifurcate into three closed loops
关Fig. 11共c兲兴 and chaos appears in the neighborhood of the
hyperbolic fixed points. However, diffusion from one sto-
兩Em − En兩2
2
,
兺
J共J − 1兲 m⬎n T共m − n兲
共22兲
where Em is the energy after the mth kick, J is the total
number of kicks. For g = 2.2 and g = 2.5, the diffusion rates
are 7.2⫻ 10−11 and 1.5⫻ 10−9, respectively.
Arnold diffusion allows the state to diffuse into higher
energy states, which destroys the quasiperiodic motion of the
quantum beating and leads to the transition to instability. As
Arnold diffusion occurs, the motion of the condensate becomes unstable and the number of the noncondensed atoms
grow exponentially, as we have seen in above discussion.
Arnold diffusion is a general property of the nonlinear
Schrödinger equation in the presence of strong interactions.
However, it may be hard to observe the whole process of
Arnold diffusion in realistic BEC experiments because of the
limited number of atoms 共106兲. As Arnold diffusion occurs,
the instability of the condensate leads to the exponential
growth of thermal atoms which destroy the condensate, as
well as invalidate the GP equation 共1兲 in a short time; while
the clear signature of the whole Arnold diffusion process
may only be observed in a relatively long period. On the
other hand, Arnold diffusion may be observed in the context
of nonlinear optics, where the GP equation 共1兲 describes the
propagation of photons. The number of photons is very large
and the interactions between them are very weak, therefore it
is possible to have a long diffusion process without invalidating the GP equation 共1兲.
C. Dynamical localized states
Although the above discussions have been focused on a
periodic state of antiresonance, the transition to instability
due to strong interactions also follows a similar path for a
dynamically localized state. The only difference is that we
start out with a quasiperiodic rather than a periodic motion in
the absence of interaction. This means that it will generally
be easier to induce instability but still requires a finite
strength of interaction.
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V. CONCLUSIONS
FIG. 12. Nonlinear effects on dynamically localized states. K
= 5, T = 1. 共a兲 Plots of average energy E共t兲 versus the number of
kicks t, where the dash-dotted line corresponds to the classical diffusion. g = 0 共dash兲, g = 1 共dot兲, g = 5 共solid兲. 共b兲 Semilog plot of the
mean number of noncondensed atoms versus the number of kicks t.
g = 1 共dot兲, g = 5 共solid兲.
In Fig. 12, we show the nonlinear effect on a dynamically
localized state at K = 5 and T = 1. For weak interactions 共g
= 1兲 the motion is quasiperiodic with slow growth in the
number of noncondensed atoms. Strong interaction 共g = 5兲
destroys the quasiperiodic motion and leads to diffusive
growth of energy, accompanied by exponential growth of
noncondensed atoms that clearly indicates the instability of
the BEC. Notice that the rate of growth in energy is much
slower than the classical diffusion rate, which means that
chaos brought back by interaction in this quantum system is
still much weaker than pure classical chaos.
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ACKNOWLEDGMENTS
We acknowledge support from the NSF and the R. A.
Welch Foundation. M.G.R. acknowledges support from the
Sid W. Richardson Foundation. J.L. was supported by the
NSF of China 共10474008兲, National Fundamental Research
Programme of China under Grant No. 2005CB3724503, and
the National High Technology Research and Development
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